Resurgence of high-energy string amplitudes

TL;DR

This paper investigates the high-energy behavior of string amplitudes using multiple mathematical approaches, revealing organization by Bernoulli numbers and applying resurgence theory to connect different kinematic regimes.

Contribution

It introduces a comprehensive framework combining saddle-point, algebraic, analytical, and geometric methods to analyze high-energy string amplitudes and their asymptotic structures.

Findings

High-energy perturbative coefficients are organized by Bernoulli numbers.

Resurgence theory transforms divergent series into transseries with non-perturbative monodromy.

Derived explicit transseries for four-point open string amplitudes.

Abstract

We analyze the fixed-angle high-energy ($\alpha' \to \infty$) structure of $n$-point tree-level string amplitudes from complementary perspectives: locally via saddle-point expansions, algebraically via difference equations and their asymptotic structure, analytically via Aomoto-Gauss-Manin connection and Mellin-Barnes representation, and geometrically via twisted intersection theory and Lefschetz thimbles. Using, in turn, saddle-point analysis and finite-difference equations in the kinematic variables, we show that the perturbative coefficients in the resulting asymptotic series in $1/\alpha'$ are organized by Bernoulli-number data, rather than by the multiple zeta values characteristic of the low-energy $\alpha' \to 0$ regime. Resurgence theory allows upgrading these divergent series to transseries whose Stokes data capture the analytic continuation between unphysical and physical kinematic regions in the form of non-perturbative monodromy contributions. We derive the transseries for four-point open string amplitudes explicitly. We also construct a differential and Mellin formulation which place their low- and high-energy expansions in a common analytic framework and unifies them as asymptotic sectors of the same underlying object. We extend the difference-equation analysis to $n \geq 5$, where it yields perturbative high-energy asymptotic expansions and leads naturally to a higher-rank connection problem. Finally, translating our asymptotic analysis into the language of twisted de Rham theory, we derive an alternative double-copy representation of the high-energy limit of closed-string amplitudes in terms of Lefschetz thimbles for any $n$.